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NON-ARISTOTELIAN  LOGIC 


BY 


HENRY  BRADFORD  SAIITH 

Assistant  Professor  of  Philosophy  in  the  University  of  Pennsylvania 


THE  COLLEGE  BOOK  STORE 

3425    WOODLAND    AVENUE 

PHILADELPHIA,  PA. 

I9I9 


NON-ARISTOTELIAN  LOGIC 


BY 


HENRY  BRADFORD  SMITH 

Assistant  Professor  oj  Philosophy  in  the  University  of  Pennsylvania 


THE  COLLEGE  BOOK  STORE 

3425    WOODLAND    AVENUE 

PHILADELPHIA,  PA. 

I9I9 


TABLE  OF  CONTENTS 
CHAPTER   I 

PAGES 

§§  1-14.  Foundations  of  the  common  logic. 
The  forms  of  immediate  inference. 
Exercises 1-13 

CHAPTER   H 

§§  15-20.     Deduction  of  the  valid  and  the  invalid 
moods  of  the  syllogism. 
Exercises 14-21 

CHAPTER   HI 

§§  21-23.     Construction  of  a  Non-Aristotelian  logic. 

Exercises 22-25 

CHAPTER   IV 

§§  24-25.     The  general  solution  of  the  sorites. 

Exercises 26-36 

CHAPTER  V 

§  26.     Alternative  systems. 

Exercises 37-40 


NON-ARISTOTELIAN    LOGIC 


CHAPTER  I 

§  I.  The  problem  of  a  science  is  to  define  the  elements, 
of  which  it  treats,  by  an  enumeration  of  their  formal 
properties.  These  properties  are  to  be  found  within  the 
system,  of  which  these  elements  are  the  parts.  The  task 
of  logic  is,  then,  to  develop  completely  its  own  system,  by 
constructing  all  the  true  and  all  the  untrue  propositions, 
into  which  its  elements  enter  exclusively. 

§  2.  The  propositions,  which  are  recognized  by  the  logi- 
cian, are: 

(i)  The  Categorical  forms,  made  up  of  terms  (represented 
in  the  proposition  x{ah)  by  the  subject  a  and  the  predicate 
b)  and  relationships  (an  adjective  of  quantity,  all,  some, 
no,  and  the  copula,  is),  viz., 

<x{ab)  =  All  a  is  all  b, 

13  (ab)  =  Some  a  is  some  b, 

y{ab)  =  All  a  is  some  b, 

t{ab)  =  No  a  is  b, 

the  word  some,  explicit  in  ^  and  y,  being  interpreted  to 
mean  some  at  least,  not  all.  This  meaning  of  the  word  will 
be  unambiguously  determined  by  the  propositions,  which 
we  say  shall  be  true  or  untrue  in  our  science.  Whenever 
we  wish  to  designate  some  one  of  the  four  forms  but  desire 
to  leave  unsettled  which  one  is  meant,  we  shall  employ  the 
notation,  x{ab),  y{ab),  z{ab),  etc. 
(2)  The  Hypothetical  forms, 

x{ab)  z  yiO'b)  =  x{ab)  implies  y(ab)  is  true, 
{x{ab)  z  y{cib)]'  =  x{ab)  implies  y(ab)  is  untrue. 


2  NoN -Aristotelian  Logic 

(3)  The  Conjunctive  form, 

x{ab)'y{ah)  =  x{ab)  and  y(ab)  are  true, 

(4)  The  Disjunctive  form, 

x{ah)  +  y{ah)  =  x{ah)  or  y{ab)  is  true. 

x{db)  is  untrue  will  be  represented  by  x'{ah),  x'{ab)  is 
untrue  by  x"{ab),  etc.  Since  in  the  proposition  x(ab)  the 
terms  are  subject,  a,  and  predicate,  b,  the  term-order  is  the 
order  subject-predicate.  Whenever  we  wish  to  leave  the 
term-order  unsettled  we  shall  place  a  comma  between  the 
terms.     Thus,  x(a,  b)  stands  for  x{ab)  or  for  x{ba). 

§  3.  A  principle,  which  is  altogether  fundamental  and 
which  will  be  taken  for  granted  at  each  step  of  our  progress, 
is  this:  If  a  proposition  is  true  in  general,  it  is  because  it 
remains  true  for  all  specific  meanings  of  the  terms  that  enter 
into  it,  although  an  untrue  proposition  may  well  enough  be- 
come true  in  the  same  circumstances.  Thus  y{ab)  z  7 (6a) 
is  untrue  in  general,  but  it  becomes  true  for  the  case,  in 
which  a  and  b  are  identical,  viz.,  y{aa)  z  y{aa).  Accord- 
ingly, when  we  write  {x{a,  b)  z  y{a>,  b)}',  we  only  assert 
that  there  is  at  least  one  value  of  a  and  one  value  of  b, 
which  will  render  x{a,  b)  z  y{a,  b)  a  false  proposition.  If 
it  has  been  established  that  x{aa)  z  yic^o)  is  untrue,  then 
we  may  at  once  infer  that  the  more  general  implication, 
x{a,  b)  z  y{a,  b),  is  untrue  as  well.  In  order  to  establish  the 
untruth  of  a  given  proposition,  it  will  be  enough  to  point  to  a 
special  instance  of  its  being  untrue. 

§  4.  In  presenting  the  materials  of  our  subject-matter  we 
shall  have  to  deal  with  two  types  of  proposition.  The 
truth  of 

{x  z  y){y  /.z)  /.  {x  z  2) 

is  independent  of  x,  y  and  z  no  matter  for  what  propositions 
X,  y  and  2  may  stand.  Such  a  general  truth  will  be  termed 
a  principle.     The  truth  of 

y{ba)y{cb)   z  y(ca) 


Foundations  of  the  Common  Logic  3 

is  independent  of  a,  b  and  c  no  matter  for  what  classes  a,  b 
and  c  may  stand.  If  such  a  general  truth  has  to  be  taken 
for  granted,  it  will  be  termed  a  postulate. 

Principles  are,  accordingly,  independent  o(  forms;  postu- 
lates are  independent  of  terms. 

§  5.  We  begin  by  setting  down  four  postulates,  the  truth 
of  which  may  be  verified  at  once  empirically  by  the  familiar 
device  of  Euler's  circular  diagrams, 

(i)  a(ba)a{bc)  z  a{ca), 

(ii)  a{]ba)^{bc)  z  /3(ca), 

(iii)  a{ba)y(cb)  z  y{ca), 

(iv)  a{ba)  i{bc)  z   ^{ca), 

and  we  shall  add  to  these, 

(v)  a'(aa)   z  o:(aa). 

This  last  assumption  illustrates  an  extension  of  the  common 
meaning  of  implication  and  is  forced  upon  us,  if  we  are  to 
allow  a{aa)  to  stand  for  a  true  proposition.  The  uses,  to 
which  such  an  extension  of  meaning  may  be  put,  will  be- 
come clear  in  the  sequel.  It  will  be  enough  to  state  that 
a  proposition,  which  is  true  for  all  meanings  of  the  terms, 
will  be  implied  by  the  proposition  whose  symbol  is  i,  and 
behaves  like  a  unit  multiplier  in  this  algebra. 

Only  a  small  number  of  the  principles,  which  we  shall 
introduce  as  necessity  requires,  are  independent  but  it  will 
not  concern  our  purpose  to  point  out  the  manner  of  their 
inter-connection. 

From  the  principle, 

{x'  z  x)  z  {y  z  x), 

we  obtain,  by  (v),  the  theorem, 

i  z  a{aa). 

By  (i),  for  a  =  b,  a{aa)a(ac)   z  a{ca),  and 

{a{aa)a{ac)   z  a(ca)}{i  z  (x{aa)}    Z   {a{ac)   Z  a(ca)}, 
by  {xy  z  z){w  z  x)  z  (^7  Z  z),  omitting  the  unit  multi- 
plier, i,  as  it  is  the  custom  to  do. 


4    ' Non-Aristotelian  Logic 

Similarly,  we  obtain 

fi(ac)  z  Pica),  from  (ii), 

yica)  z  y{ca),  from  (iii), 

e{ac)  z    i{ca),  from  (iv). 
Again, 

{a{ab)   z  a{ba)]{a{ba)   z  oc{ah)\    z   [oc{ah)   z  oc{ah)], 
by  (:x;  z  y){y  Z  z)   z  {x  Z  z). 
Accordingly, 

a(a5)   Z  a;(a&), 

^(a&)  Z  )8(a^'), 
y{ab)  z  y{ab), 
e{ab)   z   e{ab). 

A  complete  induction  of  the  members  of  this  set  and  an 
application  of  the  principle, 

(x  z  y)  z  (y  z  x'), 

yields  the  general  result, 

k'(ab)  z  k'iab),  I 

k(ab)   being  understood  as  representing  any  one  of  the 
unprimed  letters,  a,  fi,  y,  e. 

§  6.  Each  one  of  the  propositions  so  far  derived,  being 
true,  is  implied  by  the  unit  multiplier,  i.  The  contra- 
dictory (as  it  is  called)  of  i  will  be  any  proposition,  which  is 
untrue  for  all  meanings  of  the  terms  that  enter  into  it.  It 
will  be  represented  by  the  symbol,  o,  and  will  be  defined  by 

0  z  i,         {i  Z  o)', 

wherein  it  will  be  seen  that  i  stands  for  o'.     For  a  verbal 
interpretation  of  o  and  i  we  may  read : 

0  =  No  proposition  is  true, 
i  =  One  proposition  is  true. 

§  7.  The  utility  of  the  concept  of  zero,  0  (the  null- 
proposition),  and  of  one,  i  (the  <?we-proposition),  will  be 
illustrated  in  part  by  the  derivations,  which  follow.  We 
have 


Foundations  of  the  Common  Logic  5 

{a(ab)   Z  a(ab)]    £   [a{ab)a{ah)   Z  0}, 

by  (x  z  y)  Z  (xy'  z  0) ; 

{a{ab)a'{ab)   Z  o]\z   [a{ab)   z  oc"{ab)], 

by  {xy  z  0)  z  {x  z  y')-     Thus  we  should  obtain 

a{ah)  z  a"(ab), 

0(ab)  z  0"{ab), 

y{ab)  z  y"{ab), 

e(ab)  z   e"{ab), 

and,  if  we  add  to  these  the  following  assumptions, 

a"(ab)  z  oc{ab), 

fi"{ab)  z  Kab), 

y"{ab)  z  y(ab), 

e"{ab)  Z   e{ab), 

a  general  result  will  be  obvious,  viz., 

Kab)  z  k"{ab),         k"{ab)  z  Hab),  II 

wherein  the  same  restriction  is  imposed  upon  k(ab)   as 
before. 

§  8.  The  principle,  that  the  truth  of  any  one  of  the  four 
categorical  forms  implies  the  falsity  of  each  one  of  the 
others,  a  generalization,  which  will  now  be  established,  is 
characteristic  of  the  logic,  which  we  are  constructing.  We 
shall  begin  by  setting  down  the  three  characteristic  postu- 
lates, 

(vi)  fiiaa)  z  ^'(aa), 

(vii)  y{aa)  z  y'{aa), 

(viii)  e{aa)   z    e'{aa). 

Then,  by  the  principle, 

(x  z  x')  z  (x  z  y), 

we  may  establish  at  once, 

fi(aa)  z  0, 

y{aa)  z  0, 

i{aa)  z  0. 


6  Non-Aristotelian  Logic 

Postulate  (ii)  above  yields,  for  a  =  c,a{ba)^{ha)  z  I3(aa); 
{a{ba)^{ba)   z  fi{aa)\[P{aa)   I  o]    z.   \a{ba)^{ba)   z  o], 
by  {x  Z  y)(y  Z  z)   z  {x  z  z); 

{a(ba)l3{ba)   Z  o\    z   {a{ba)   z  fi'{ba)\, 

by  {xy  z  o)   z  {x  z  y'). 

Similarly,  by  (iii),  a{ba)   z  y'(ab); 

{a(ab)   z  a{ba)}{a(ba)   Z  y'(ab)}    z   Mab)   Z  y'{ab)}, 

by    (.r  z  y)(y  Z  z)   z  (x  z  z);    and,    by    (iv),    we   obtain 
a{ab)   Z  e'{ab). 

If  now  we  postulate, 

(ix)  y(ba)y(cb)   z  y{ca), 

(x)  e(ba)y(cb)   z   e{ca), 

there  result  as  before  €(ab)   z  y'(ab)  and  y(ab)   z  y'{ba), 

and  if 

(xi)  Kab)  z  y'(ab), 

(xii)  (3(ab)   z   e'iab), 

all  of  the  remaining  implications  of  this  form,  viz., 

e(ab)  z  a'(ab),  y{ab)  z  a'(ab), 
e{ab)  z  ^'{ab),  y{ab)  z  I3'{ab), 
e[ab)   z  y'{ab),         I3{ab)   z  oi'{ab), 

are  obtained  at  once  from  those  that  have  just  been  estab- 
lished by 

{x  z  y')  z  (y  z  x'). 

If,  now,  k(ab)  and  wiab)  can  not  represent  the  same 
categorical  form,  y(ab)  and  y{ba)  being  considered  distinct, 
and  if  further  k{ab)  and  'w{ab)  can  stand  only  for  the  un- 
primed  letters,  a,  (3,  y,  e, 

kiab)  z  w'{ab).  Ill 

This  generalization  is  one  of  the  characteristics,  which 
marks  an  equivalence  of  the  logic,  whose  foundations  are 
here  set  down,  with  the  classical  logic  of  Aristotle.  If  we 
were  to  follow  an  accepted  modern  tradition,  which  regards 
e{aa')    as    a    true    proposition,  (a'  =  non-a),    not    all  im- 


Foundations  of  the  Common  Logic  7 

plications  of  this  type  will  hold,  for  y{ab)  z  ^'{ab)  and 
t{ab)  z  y'iab)  become  €(ot)  /_  y'{oi)  and  y{oi)  z  ^'{oi),  for 
a  =  0,  b  =  i  (see  §  13  below).  In  such  a  logic,  which  is,  in- 
deed, an  alternative,  or  Non-Aristotelian  system,  but  which 
gives  up  the  advantage  gained  by  our  symmetry,  we  should 
have  to  write  {y{ab)  z  e'(ab)}'  and  {e{ab)  z  y'iab)}'. 
Moreover,  postulate  (x),  from  which  these  results  are 
derived  becomes  itself  untrue  and  the  same  remark  applies 
to  e(ab)y{cb)  z  e(ca)  and  y{ab)e(c,  b)  z  ^(ca),  implications, 
which  will  later  be  made  to  depend  on  postulate  (x). 

§  9.  We  shall  now  establish  the  untruth  of  certain  forms 
of  implication,  making  them  ultimately  depend  upon  the 
invalidity  of  i  z  0,  whose  untruth  is  set  down  as  a  matter 
of  definition. 

Suppose  a{aa)  z  ci'{aa)  were  true. 

[i  z  oi{aa)]   z   [a'iaa)   z  0}, 

by  (x'  Z  y)   Z  {y'  Z  X) ; 

[i  z  a{aa)\{a{aa)   z  ociaa)]   z  [i  Z  a'iad)], 

by  {x  Z  y){y  Z  z)  z  {x  z  z)\ 

[i  z  ci'{aa)][oc'{aa)  Z  0]   z  {i  Z  0}, 

by  the  same  principle. 
But  i  z  0  is  untrue. 

.".     a{aa)  z  a'(aa)  is  untrue. 
Again, 

{(x(aa)  z  oc'(aa)}'{fi(aa)  z  a'(aa)}  z  {oc{aa)  z  /3(aa)}', 
by  (x  z  z)'(y  z  z)   z  (x  Z  y)'; 

{a{aa)  Z  ^' (aa) }  {a{aa)  z  /3M  }'  Z  {P'(aa)  z  i^M }', 
by  (x  z  y)ix  z  z)'  z  (y  Z  z)'. 

Accordingly,  we  have 

{a  (aa)  z  a'(aa)}'j 

{^'(aa)  z  ^(aa)}', 

{y'(aa)  z  y  (aa)}', 

{  t'{aa)  z   e  {ad)]'. 


8  Non-Aristotelian  Logic 

and,  since  the  untruth  of  any  proposition  is  implied,  when- 
ever we  can  point  to  a  special  instance  of  its  being  untrue, 

it  follows  that 

{a  {ah)   Z  a{ah)]\ 

{fi'iah)  z  Kah)V, 
{y'{ah)  z  y{ab)Y, 
{,'{ab)   z    e{ab)V, 

The  first  and  third  members  of  the  set 

{a' {ah)  z  a{ah)]\ 

{^{ah)  z  ^'{ah)Y, 

{y{ab)  Ly'{ah)Y, 

[e{ah)  z   e{ah)Y, 

will  be  established  on  making  a  =  o,  h  =  t.  For  the  reduc- 
tion of  the  second  and  fourth  see  exercise  (lo)  at  the  end 
of  this  chapter. 

As  a  result  of  a  complete  induction  of  the  members  of 
these  sets  and  upon  application  of  {x  /.  y)'  Z  {y'  Z  x')', 
it  follows  that 

{k  {ah)  z  k'  {ah)}',         {y  {ah)  z  k  {ab)\',  IV 

{k'{ab)  z  k"{ah)}',         {k"{ah)  z  k'{ah)Y. 

§  10.  If  a' {ah)  z  fi{cih)  were  a  true  implication,  we  should 
have: 

{y{ah)   z  a' {ah)}  {a' {ah)   z  Kah)}   Z   \y{ah)   z  ^{ah)], 
by  {x  z  y){y  z  z)  z  {x  z  z)\ 

{y{ah)  z  (3{ah)}{Kah)  z  y'{ah)}   z  {y{ah)  z  y'{ah)}, 
by  the  same  principle. 

.*.     a  {ah)  z  I3{ah)  is  untrue. 

Applying  the  same  method  of  reduction  there  will  result : 

{a' {ah)  z  Kah)Y,  {^'{ah)   Z  y{ah)Y, 

{a' {ah)   Zy{ah)}\  {13' {ah)   z    e{ah)}', 

{a'{ah)   z   e{ah)}',  {y'{ah)   z    e{ah)\\ 

{y'{ah)  zy{ha)Y, 

and  upon  application  of 


Foundations  of  the  Common  Logic  9 

(x'  /  y)'  /  (y'  /.  oc)' 

{e'{ab)   z  a{ab)]',  {y'{ab)  \a{ab)V, 

{e'{ab)   z  Kab)V,  \y'{ab)   z  Kab)]', 

W{ab)   z  y{ab)\\  {(3'(ab)   z  a(ab)}'. 

We  are  now  prepared  to  lay  down  the  final  generaliza- 
tions which  are  given  below.  From  the  propositions  that 
have  just  been  enumerated  there  will  follow 

{w'{ab)   z  k(ab)\',  V 

from  III  and  V,  by 

{x  z  y)  z  (y'  z  x'), 
(x  z  y)'  z  iy'  z  x')', 
k"{ab)  zw'iab),         {w'iab)  zk"{ab)Y,  VI 

from  III  and  IV,  by 

{x  Z  z)'{y  z  z)   z  (x  z  y)', 
(x  z  'v)'  z  (y'  z  x')' 
{k(ab)  zw"{ab)Y,         {w"{ab)  z  k(ab)}'.         VII 

§  II.  In  order  to  classify  the  categorical  forms  under  the 
heads,  contradictories,  contraries,  subcontraries,  and  sub- 
alterns, let  us  consider  what  special  meanings  of  x(ab)  and 
y(ab)  render  true  or  untrue, 

(i)  x(ab)  z  y'icib), 

(2)  y'{cib)   z  x{ab). 

If  x{ab)  and  y{ab)  satisfy  (i)  and  (2)  together,  x{ab)  is 
said  to  be  contradictory  to  y{ab).  By  I,  k'{ab)  is  contra- 
dictory to  k{ab)  and,  by  II,  k{ab)  is  contradictory  to  k'{ab). 

If  x{ab)  and  y{ab)  satisfy  (i)  alone,  x{ab)  is  said  to  be 
contrary  to  y{ab).     By  III  and  V,  k{ab)  is  contrary  to  w{ab). 

If  x{ab)  and  y{ab)  satisfy  (2)  alone  x{ab)  is  said  to  be  sub- 
contrary  to  y{ab).     By  VI,  k'{ab)  is  subcontrary  to  w'(ab). 

If  x{ab)  and  3'(a^)  satisfy  neither  (i)  nor  (2),  .T(a^)  is 
said  to  be  subaltern  to  y{ab).  By  VII,  ^(a6)  is  subaltern 
to  w'{ab),  and,  by  IV,  k{ab)  and  ^'(a^)  are  each  the  sub- 
alterns of  themselves. 


10  Non-Aristotelian  Logic 

§  12.  In  order  to  classify  terms  under  the  heads,  contra- 
dictories, contraries,  suhcontraries  and  subalterns,  let  us  con- 
sider what  special  meanings  of  a  and  b  render  true  or  untrue, 

(1)  a'iab')   z  y(ab'), 

(2)  a'ib'a)   z  yib'a). 

The  postulate  a' (a 'a')  z  a{a'a')  imp\iesa'{a'a')  /.y{a'a'), 
for 

[a'ia'a')   A  a{a'a')\    z   [a{a'a')   Z.  a"{a'a')\, 

by  {x  z  y)  Z  {y'  Z  x') ; 

{a'ia'a')   z  oi"{a'a')]   z   W{a'a')   z  y{a'a')], 

by  (x  z  x')   z  {x  z  y). 

If  contradictory  terms  be  those  meanings  of  a  and  b  that 
render  (i)  and  (2)  true  together,  then,  by  a  {a' a')  z  y{a'a'), 
it  follows  that  a  and  a'  are  contradictory. 

If  contrary  terms  be  those  meanings  of  a  and  b  that  cause 
(i)  alone  to  become  true,  then  if  we  assume, 

a'ioi)   z  y{oi), 
[a'iio)   z  y{io)\', 

where  0'  =  i,  i'  =  0,  we  derive  in  particular  the  fact  that 
o  is  the  contrary  of  itself. 

If  subcontrary  terms  be  those  meanings  of  a  and  b  that 
cause  (2)  alone  to  become  true,  then,  from  the  assumptions 
just  written  down,  it  follows  in  particular  that  i  is  the 
subcontrary  of  itself. 

If  (i)  and  (2)  remain  untrue  for  some  special  meaning 
of  a  and  b,  then  a  is  said  to  be  the  subaltern  of  b.  From  the 
two  equivalent  propositions, 

{a'{aa')   z  y{aa')]',  {a{a'a)   z  y{a'a)\', 

whose  untruth  may  be  established  on  making  a  =  i,  a'  =  0, 
in  the  first,  a'  =  i,  a  =  0,  in  the  second,  and  which  imply 

{a'iab')  zy(ab')}',  {a'ib'a)   Z  yib'a)}', 

it  will  be  seen  that  a  is  in  general  the  subaltern  of  b  and 
of  itself. 


Foundations  of  the  Common  Logic  ii 

§  13.  If  the  meaning  of  zero  (0)  is  unique;  that  is,  if 
we  assume, 

i  z   {a  (to)   z  yito)}', 

which  is  the  same  as, 

{a  (to)   Z  7 («*<?)}    Z  0, 
we  should  have, 

a{io)   z  0,         y{io)   z  0, 
and  from 

a{oi)   z  y{oi), 
oi{oi)  z  0,        y'{oi)  z  0. 

These  last  results  and  others,  that  have  gone  before 
(viz..  Ill),  yield: 

a  {00)  Z,  0,  a  {oi)  Z  0,  a{io)  Z  0,  a{ii)  Z  0, 

/3  {00)  z  0,  /3  (oi)  z  0,  fiiio)  z  0,  j8  {ii)  z  0, 

y  (oo)  z  0,  y'{oi)  Z  0,  y{io)  z  0,  y  {ii)  z  0, 

€  {00)  z  0,  €  {oi)  z  0,  e{io)  z  0,  e  {ii)  z  0. 

We  may  now  establish 

{a'{aa')   z  a{aa')Y ,  U'{aa')   z   e{aa')Y, 

{y'{aa')  z  y{aa')Y,         {y{aa')  z  y'{aa')Y, 

and  if  we  postulate 

(xiii)  a{aa')  z  a{aa'), 

(xiv)  ^{aa')  z  fi'{aa'), 

(xv)  {e{aa')  Ze{aa')Y, 

(xvi)  {y'{aa')  z  y{a'a)Y, 

the  truth  or  untruth  of  every  remaining  variety  of  imme- 
diate inference,  x{a,  b)  Z  y{ci,  b),  may  be  derived. 

§  14.  The  operation  of  simple  conversion  consists  in  the 
interchange  of  subject  and  predicate.  From  y{io)  z  0  and 
y'{oi)  z  0,  which  are  imposed  upon  us  by  the  definition 
of  the  null-class  (c)*,  it  will  appear  that  the  inconvertibility 
of  7  is  fundamental;  for 

[i  z  oY{y{io)  z  0}  z  [i  Z  y{io)Y 

*  The  null-class  (o)  is  to  be  understood  as  the  class  that  contains  no  objects, 
or  none  of  the  objects  that  are  in  question.  The  universe  (»)  is  the  class  that 
contains  all  of  the  objects  that  are  in  question. 


12  Non-Aristotelian  Logic 

by  (x  A  zy^y  Z  z)  Z  (x  z  y)',  and 

{i  Z  y(oi)]{i  z  y(io)}'  z  {y{oi)  z  yiio)}' 

by  (x  z  y)(x  z  z)'  z  {y  Z  z)'. 

.'.     y(ab)  z  y{ba)  is  untrue. 

Exercises 
I.  The     meaning  of  logical  equality  is  given  by 

(x  z  y)(y  z  x)  z  {x  =  y), 
{x  =  y)  z  (x  z  y){y  z  x). 

If  k{ab)  =  k{ab)k{ab)  and  k(ab)  z  w'(ab),  show  that 

a{ab)  z  ^'{ab)Y(ab)e'{ab)y'(ba), 

I3{ab)  Z  cc'{abW{ab)  e'{ab)y'iba), 

y{ab)  Z  a'(ab)l3'{ab)e'{ab)Y{ba), 

eiab)  Z  a'{ab)^'(ab)y'{ab)y'(ba), 


by  the  aid  of 


2.  If 


(.X  z  y)iy  z  z)  z  (x  z  z), 

(x  Z  y)  Z  {zx  Z  zy). 

^'{ab)y'{ab)t'{ab)y'{ba)  z  a{ab), 

a'{ab)y'{ab)i'{ab)y'{ba)  Z  ^{ab), 

a'(ab)^'(ab)e'(ab)y'{ba)  Z  y{ab), 

a'{ab)^'{ab)y'{ab)y'{ba)  Z  eiab), 


establish 


a'iab)  =  fiiab)  +  y{ab)  +  c(aZ?)  +  y{ba), 

^'{ab)  =  a{ab)  +  y{ab)  +  e{ab)  +  y{ba), 

y'{ab)  =  oc{ab)  +  ^{ab)  +  t^ab)  +  y{ba), 

t'iab)  =  a{ab)  +  fiiab)  +  y{ab)  +  y{))a), 

assuming  that  the  contradictory  of  a  product  is  the  sum  of  the 
contradictories  of  the  separate  factors  and  assuming  the  right 
to  substitlite  k{ab)  directly  for  k"{ab). 

3.  Assuming  x{ab)  =  x{ab)x(ab),  x(ab)  =  x{ab)  +  x(ab),  show 
that 

a'{ab)^'{ab)  =  y{ab)  +  e{ab)  +  y{ba),  etc.,  etc., 
a{ab)  +  ^{ab)  =  y'(ab)e'(ab)y'(ba),  etc.,  etc. 


Foundations  of  the  Common  Logic  13 

4.  Establish  the  general  results, 

kiab)  =  k(ab)w'{ab),        k'{ab)  =  k'iab)  +  w(ab), 
k'iab)'w{ab)  =  0, 

5.  From  the  principle,  ix  /  z)'{y  Z  2)  z  (^  Z  3')',  and  the 
postulate,  {a{aa)  Z  a'(aa)}',  derive 

{oi{aa)  z  &{aa)Y,         {a{aa)  z  y{aa)Y ,         {a{aa)  Z  f^{aa)Y . 

6.  By  the  aid  of  the  principles, 

{x  z  y){y  Lz)  /_  {x  z  z),       (x  z  x')  A  ix  z.  y), 

from  the  postulate,  a'{aa)  Z  a{aa),  and  results  already  estab- 
lished, (viz.,  Ill),  show  that  all  propositions  of  the  form, 
x{aa)  Z  yicLo),  except  the  three  cases  in  the  last  example,  are 
true  implications,  x{aa)  and  y{aa)  representing  only  the  un- 
primed  letters. 

7.  Show  by  the  method  of  the  last  example  that  a  (aa)  z  a{aa) 
is  the  only  untrue  implication  of  the  form  x{aa)  Z  y'{aa). 

8.  Derive  seven  true  implications  of  the  form,  x'{aa)  Z  yiao), 
and  nine  untrue  implications  of  the  same  form. 

9.  Establish  the  untruth  of 

k{a,  b)  z  w(a,  b),         k'{a,  b)  Z  w'ifl,  b). 

10.  Establish  the  untruth  of  /3(a6)  Z  ^'{ab)  and  e{ab)  Z  e'(a6) 
by  making  them  depend  upon  the  untruth  of /3(6a)j8(c&)  Z  ^'{ca) 
and  €iba)e{cb)  Z  i'(ca)  respectively  (see  the  postulates  of  the 
next  chapter).     Thps, 

{^{ba)^(cb)  Z  )8'(ca)}'  Z  {^{ba)Kcb)Kca)  Z  0}', 
by  (x  z  y)'  Z  (^3'  Z  0)'; 
{^{ba)^{cb)^{ca)  Z  o}'{/3(<:6)/3(ca)-o  Z  0} 

Z  {^{ba)^{cb)^{ca)  Z  |S(c6))8(co)-o}', 
by  (xy  z  z)'(w  /.  z)  /H  {xy  z  w)'; 

{0{ba)Kcb)Kca)  Z  i8(c&)^(ca)-or  Z  {^(M  Z  0}' 
by  (x2  Z  2^)'  Z  (^  Z  y)'; 

{^(M  Zo}'  A  {^(ba)  Z  i3'(M}', 
by  (:c  Z  0)'  z  (^  Z  x')'. 


CHAPTER   II 

§  15.  A  syllogism  is  an  implication  belonging  to  one  of 
the  types, 

1.  x{ha)y{cb)  z  2(ca), 

2.  x(ab)y{cb)  z  z{ca), 

3.  x{ba)y{bc)  z  z{ca), 

4.  x{ab)y{bc)  z  z{ca). 

These  differences  are  known  as  the  first,  second,  third 
and  fourth  figures  of  the  syllogism  respectively.  The  two 
forms  conjoined  to  the  left  of  the  implication  sign  are  called 
the  premises  and  the  form  to  the  right  of  the  implication 
sign  is  called  the  conclusion.  The  predicate  of  the  con- 
clusion is  called  the  major  term  and  points  out  the  major 
premise,  which  by  convention  is  written  first,  while  the 
subject  of  the  conclusion  is  called  the  minor  term  and 
points  out  the  minor  premise.  The  term,  which  is  common 
to  the  premises  and  which  does  not  appear  in  the  conclusion, 
is  called  the  middle  term. 

Since  x,  y  and  z  may  assume  any  one  of  the  four  values, 
a,  /3,  7,  e,  there  will  be  sixty-four  ways  in  each  figure,  called 
the  moods  of  the  syllogism,  in  which  xy  /.  z  may  be  ex- 
pressed. True  syllogistic  variants  are  called  valid  moods. 
Those  that  are  untrue  are  called  invalid  moods. 

It  will  be  convenient  to  deduce  in  the  first  place  all  of 
the  valid  moods  of  syllogistic  form  that  exist  and  to  estab- 
lish later  on  the  invalidity  of  those  moods  that  remain. 
In  what  follows  we  shall  suppose  that  x,  y  and  z  stand 
only  for  the  unprimed  letters.  Thus:  we  shall  refer  to 
x{a,  b)y(b,  c)  z  z{ca),  x'(a,  b)y{b,  c)   z  z'{ca), 

x(a,  b)y{b,  c)   z  z'{ca),  etc., 

as  specific  types. 

14 


Moods  of  the  Syllogism  15 

§  16.  The  valid  moods  of  the  syllogism, 

x(a,  b)y{b,  c)  /_  z{ca), 

twenty-nine  in  all,  which  are  not  set  down  among  the 
assumptions  of  §  5  and  §  8,  may  be  derived  at  once  by  the 
following  principles  : 

{xy  z  z){'w  z  x)   z  ("wy  z  z), 

(x  z  y){y  z  z)  z  (x  z  z), 

{xy  z  z)   z  {yx  z  z). 

Thus,  from  postulate  (x),  by  the  second  principle, 
{e{ba)y{cb)  z  e(m) }  {e(ca)  z  e(ac)}  /  {e{ba)y{cb)  z  e(ac)}, 
and,  since  the  term-order  in  the  conclusion  is  now  reversed, 
so  that  the  major  term  has  become  the  minor  term  and 
the  minor  term  has  become  the  major  term,  it  will  be 
necessary  to  employ  the  third  principle  to  restore  the 
normal  order  of  the  premises.     Accordingly, 

{(:{ba)y{cb)  z  6(ac) }    z  {y(cb)e(ba)   z  e(ac)}, 

and  it  will  be  seen  that  the  term  order  in  this  result  is  that 
of  the  fourth  figure.  The  second  principle  (above)  thus 
enables  us  to  convert  simply  in  the  conclusion  and  the  effect 
of  simple  conversion  in  the  conclusion  is  to  change  the  first 
figure  to  the  fourth. 

Similarly,  since  the  third  principle  enables  us  to  arrange 
the  premises  in  either  order,  the  first  principle  will  allow  us 
to  convert  simply  in  either  premise,  if  that  premise  be  not 
in  the  7-form.     Thus,  from  postulate  (i)  of  §  5, 

{a{ba)(x{bc)   z  a{ca)]    z  {a{bc)a{ba)   z  a(ca)}, 

by  the  third  principle  (above) ; 

{a(bc)a(ba)   Z  (x(ca)}  {a(cb)   Z  a(bc)} 

Z   {a{cb)a{ba)   z  ci{ca)l^ 
by  the  first  principle  (above) ; 

{a{cb)a{ba)   z  a{ca)]    z   {a(ba)a(cb)   z  (x{ca)}, 

by  the  third  principle  (above) ;  and  this  result  is  a  valid 
mood  of  the  first  figure.      However,   when  it  is  desired 


1 6  NoN- Aristotelian  Logic 

to  convert  simply  In  the  minor  premise,  it  will  be  more  con- 
venient to  employ  at  once  the  principle, 

(xy  A  z){w  z  j)   Z  {xw  z  z), 

and  avoid  two  of  the  three  steps,  that  would  otherwise  be 
necessary. 

Exercise 

From  postulates  (i-iv)  of  §  5  and  postulate  (x)  of  §  8  derive 
twenty-three  valid  moods  of  the  syllogism  by  the  aid  of  the 
principles, 

{xy  Z  z){w  Z  x)  Z  (wy  Z  2), 

(^3;  Z  z){w  Z  y)  Z  (xiv  Z  z), 

(xy  Z  z)(z  z  w)  z  (xy  Z  w), 

(xy  Z  z)  z  {yx  Z  z). 

§  17.  The  valid  moods  of  the  syllogism, 

x{a,  b)y{h,  c)  z  z'{ca), 

one  hundred  and  forty- two  in  number,  as  well  as  those  of  the 
syllogisms,  x{a,  b)y'{b,  c)  z  z'ica)  and  x'{a,  h)y{b,  c)  z  z'{ca), 
which  number  thirty-one  and  twenty-seven  respectively, 
may  now  be  obtained  from  the  results  of  §  16  and  the  forms 
of  immediate  inference  contained  in  §  8,  by  the  aid  of  the 
additional  principles, 

{xy  z  z')   z  {xz  z  y'), 
{xy  z  z')   z  {zy  z  x'). 

The  examples,  which  follow,  will  be  enough  to  illustrate 
the  method. 

(i)         \^{bd)^{cb)   z  tM}   z   {i{ba)^'{ca)   z  l'{cb)\, 

by  {xy  z  z)   z  [xz'  z  y'). 

(2)  {y{abW{cb)   z  y'{ca)}[e{cb)   z  y'{cb)] 

Z  h{abHcb)  zy'ica)], 
by  {xy  z  z){w  z  y)   Z  {xw  z  z). 

(3)  [y{ab),{cb)  z  y'{ca)]   z  [y{ab)y{ca)  z  e'{cb)\ 
by  {xy  z  z')   z  {xz  z  y')- 


Moods  of  the  Syllogism  17 

No  other  valid  moods  of  syllogistic  form  exist,  except 
those  that  have  now  been  enumerated,  as  will  appear  in 
the  sequel,  when  all  of  the  remaining  variants  shall  have 
been  declared  untrue. 

Exercises 

1.  From  postulate  (x)  of  §  8  above  deduce  six  valid  implica- 
tions of  the  form,  x{a,  b)y'{b,  c)  /.  z'{ca). 

2.  From  postulate  (x)  of  §  8  above  deduce  thirty-three  valid 
implications  of  the  form,  x{a,  b)y{b,  c)  Z  z'{ca). 

3.  Assuming  the  special  conditions  mentioned  at  the  end  of 
§  8  to  hold  true,  show  that  postulate  (ix)  of  §  8  yields  only  thirteen 
valid  implications  of  the  form  given  in  the  last  exercise. 

§  18.  It  will  be  convenient  in  establishing  the  invalid 
moods  of  the  syllogism  to  begin  with  the  form 

x(a,  h)y{b,  c)  z  z'{ca). 

Any  invalid  mood  under  this  head,  which  contains  an 
a-premise  or  an  a-conclusion,  may  be  shown  to  be  in^-alid 
by  identifying  terms  in  the  a-form.     Thus: 

1.  Suppose  a(ba)y(cb)  z  y'{ca)  were  valid,  and  identify 
terms  in  the  major  premise. 

{a{aa)y{ca)  z  y'{ca)]{i  z  oi{aa)]   z  {y{ca)  z  y'{ca)}, 

by  {xy  /.  z)(w  z  x)  z  ("wy  z  z). 

.'.     a{ba)y{cb)   z  y'{ca)  is  invalid. 

2.  Suppose  y{ab)y{cb)  z  oi'ica)  were  valid  and  identify 
terms  in  the  conclusion. 

{y{ab)y{ab)   z  a' {aa)  ]  [a  {aa)   z  0}    z  (y{ab)y(ab)   z  0}, 

by  (xy  Z  z)(z  z  w)   z  {xy  Z  iv) ; 

{y{ab)y(ab)   z  0}    z  {y(ab)   z  y'iab)}, 

by  {xy  z  0)   z  {x  Z  y'). 

.'.     y{ab)y{cb)  z  oi'{ca)  is  invalid. 


i8 


Non-Aristotelian  Logic 


Exercise 

Establish  the  invalidity  of  the  thirty-four  moods  of  the  syl- 
logism, x(a,  h)y{h,  c)  £  z'ica),  which  are  invalid  and  which 
contain  an  a-premise  or  an  a-conclusion. 

In  order  to  deduce  the  invalid  moods,  which  remain, 
eighty  in  all,  it  will  be  necessary  to  add  eleven  postulates 
to  the  ones  already  set  down.     These  assumptions  are: 


(xvii) 

{Kha)Kch)   Z  ^'{ca)Y, 

(xviii) 

'P(ba)0(cb)  z  y'{ca)y, 

(xix) 

Kba)l3icb)   z   e'ica)}', 

(xx) 

y{ab)y{cb)   z  fi'{ca)}\ 

(xxi) 

y(ba)y(bc)   z  fi'ica)}', 

(xxii) 

y{ba)y{cb)   z  y'{ca)]', 

(xxiii) 

y{ab)y{cb)   z   ^'{ca)]', 

(xxiv) 

\<ba)y{bc)   z  ^'{ca)]\ 

(xxv) 

U{ba)y{bc)   z    €'(m)}', 

(xxvi) 

e{ba)e{cb)   Z  /3'M}', 

(xxvii) 

'e{ba)e{cb)   z   e{ca)]'. 

Exercise 

From  postulates  (xvii-xxvii)  deduce  sixty-nine  other  non- 
implications  of  the  same  form,  by  the  aid  of  the  additional 
principles, 

{xy  z  z)'{w  /.  z)  z.  {xy  Z  w)' 

{xy  Z  zy{x  Z  w)  Z  ("d'y  Z  z)' 

{xy  Z  z)'(y  Z  w)  Z  {xw  Z  z)' 

{xy  Zz')'  Z  {xz  z  y'Y 

{xy  zz')'  z  {zy  Z  x')' 

{xy  Z  z)'  Z  {yx  Z  z)' 

§  19.  All  of  the  invalid  moods  of  the  syllogistic  form, 
x{ay  b)y{b,  c)   z  z{ca),  x'{a,  b)y{b,  c)   z  z'{ca)  and 

x{a,  b)y'{b,  c)   z  z'{ca), 

may  be  deduced  at  once  from  the  results  that  have  now 
been  established.  A  few  examples  will  be  enough  to 
illustrate  the  method. 


Moods  of  the  Syllogism  19 

U(ba)y{bc)  z  0'{ca)y  z   {I3{ca)e{ba)   z  y'(bc)}\ 

by  (xy  A  z'Y  z  {zx  z  y')' \ 

{KabHcb)  zY{ca)y{Kab)   ^  y'iab)] 

Z   {y'iab)e(cb)   ZT'Mr, 
by  {xy  z  2)'(-^  Z  ^)   Z  (^3'  Z  s)'; 

{7'(a6)6(c^;)   Zt'M'}    Z   {€(c^')7M    ZtC^^*)}/ 
by  Ct'^/  z  z')'  z  (>'2  Z  :>^)'; 

{7'(a6)6(c6)    Z  7'(^a)r   Z   (tMt'C^^)    Z  e'icb)}', 
by  Gtj'  z  2')'  Z  (zx  z  3'')'; 

{e(^^«)7(^'c)   z  fi'{ca)y{eica)   z  /3'M } 

Z   {e(6a)7(^c)   Z  e(ca)}', 

by  (xj  z  z)'(w  z  z)  z  (xy  z  iv)'; 

{e{ba)y{bc)  z  ^{ca)]' \eiac)  Z^ica)]   z  {y{bc)e{ba)  Ze{ac)}\ 

by  {xy  z  z)'{w  Z  z)  z  {yx  z  "w)'. 

Exercises 

1.  Show  that  there  exist  no  valid  implications  of  the  form, 
x'{a,  b)y{b,  c)  Z  z{ca)  ov  x{a,  b)y'{b,  c)  Z  z{ca),  and  consequently 
noneof  theform,:x;'(a,  6)y(6,  c)  Z  z{ca)  or  x' {a,  b)y'{b,c)  Z  z'{ca). 

2.  Show  that  as  a  result  of  a  complete  induction  of  the  moods 
in  question,  (a)  a  valid  mood  of  the  syllogism,  whose  premises 
and  conclusion  are  all  unprimed  forms  and  one  of  whose  premises 
is  of  the  same  form  as  the  conclusion,  will  remain  valid,  when 
the  other  premise  is  put  in  the  a-form ;  and  (b)  a  valid  mood  of 
the  syllogism,  whose  premises  are  unprimed  forms  and  whose 
conclusion  is  a  primed  form  and  one  of  whose  premises  is  of  a 
different  form  from  the  conclusion,  will  remain  valid,  when  the 
other  premise  is  put  in  the  a-form. 

§  20.  It  will  be  well  at  this  point  to  Indicate  the  equiva- 
lence of  the  logic,  whose  system  has  now  been  partially 
developed,  with  the  classical  science,  perfected  in  the 
Organon  of  Aristotle. 

The  four  categorical  forms  employed  by  the  traditional 
logic  and  denoted  by  the  letters,  A,  E,  I,  0,  are: 


20  Non-Aristotelian  Logic 

A(ab)  =  All  a  Is  b, 
E(ab)  =  No  a  is  &, 
I(ab)  =  Some  a  is  b, 
0(ab)  =  Some  a  is  not  b, 

the  word  some,  which  is  expressed  before  the  subject  of  / 
and  0  and  understood  before  the  predicate  of  A  and  /, 
being  interpreted  to  mean,  some  at  least,  possibly  all. 

This  set  of  four  forms  satisfies  certain  conditions,  which 
are  characteristic  of  the  system,  i.e., 

1.  Corresponding  to  each  member  of  the  set,  there  is 
another,  which  stands  for  its  contradictory; 

2.  The  relation  of  subalternation,  A(ab)  implies  I(ab), 
holds  true ; 

3.  A(ab)  becomes  true,  when  subject  and  predicate  have 
been  identified ; 

4.  The  subject  and  predicate  of  E{ab)  and  I{ab)  alone  are 
simply  convertible. 

We  should  have  to  have, 
to  satisfy  (i),  A{ab)0{ab)  a  0, 

E{ab)I{ab)  z  0, 

A'{ab)0'{ab)  z  0, 

E'{ab)r{ab)  z  o', 
to  satisfy  (2),  A{ab)E{ab)  z  0; 

to  satisfy  (3),  A'{aa)  z  0, 

to  satisfy  (4),  I{ab)   z  liba).'^ 

Today  it  is  all  but  universally  taken  for  granted  that 
not  all  of  these  conditions  hold  true,  if  the  terms  are  allowed 
to  take  on  the  meanings  nothing  and  universe,  and  it  is 
usual  to  retain  (i),  (3)  and  (4)  and  to  assert  that  the  relation 
of  subalternation  of  the  classical  logic  is  false.     This  modern 

*  It  would  in  the  end  economize  assumptions  to  take  E{ab)A{ch)  Z,  E{ca) 
for  granted  instead  of  this  last  form  of  immediate  inference,  for 

E{ab)A{ch)  z  E{ca)  yields  E{ab)A{bb)  z  E{ba),  for  b  =  c; 
\E{ab)A{bb)  Z  E{ba)\{i  z  A{bb)}   Z  {E{ab)  z  E{ba)],  by  (3); 
[E{ab)  z  E{ba)]   Z  {E'{ba)  z  E'{ab)\; 

{I{ba)  Z  E'{ba)]{E'iba)  z  E'{ab)]   Z  {I{ba)  Z  E'{ab)],  by  (i); 
{I{ba)  z  E'{ab)][E'{ab)  z  liab)]  z  {I{ba)  z  I{ab)],  by  (i). 


Moods  of  the  Syllogism  21 

tradition  is,  however,  based  upon  a  misapprehension,  upon 
the  supposed  necessity  of  retaining  A(ab)  =  E{ab').  The 
impHcations  set  down  in  the  following  table  will  be  seen 
to  satisfy  not  only  conditions  (1-4)  but  also  the  definition 
of  the  null-class  and  the  consequences,  that  follow  from 
that  definition.     These  implications  are: 


A' {00)  z  0, 

A'{oi)  z  0, 

A  {to)   z  0, 

A'iii)   z  0, 

E(oo)  z  0, 

E{oi)  z  0, 

E{io)   z  0, 

E{ii)  z  0, 

I  {00)  z  0, 

I  {oi)   Z  0, 

I'{io)   Z  0, 

I  {ii)  Z  0, 

0(oo)  z  0, 

0{oi)   z  0, 

O'iio)   z  0, 

0{ii)   z  0. 

We  may  now  state  the  connection  between  the  traditional 
propositions,  A,  E,  I,  0,  and  the  special  categorical  forms, 
which  have  been  employed  in  the  text.  This  connection 
is  expressed  by  the  following  equalities : 

A(ab)  =  a(ab)  +  yiab), 

E{ab)   =  e{ab), 

I{ab)  =  a{ab)  +  K^b)  +  y{ab)  +  y{ba), 

0{ab)  =  e{ab)  +  /3(a6)  +  yiba). 

and  it  will  be  readily  seen  that  these  values  of  ^,  E,  I 
and  0  satisfy  conditions  (1-4). 

If,  now,  we  express  a,  13,  7  and  e  in  the  members  of  the 
set,  A,  E,  I,  0,  we  should  have 

a{ab)  =  A{ab)A{ba), 
^(ab)  =   l(ab)0(ab)0(ba), 
y(ab)  =  Aiab)0{ba), 
i{ab)  =  E{ab), 

which  may  be  verified  by  actually  multiplying  out  these 
products  as  in  ordinary  algebra  and  allowing  the  product 
k{ab)w{ab)  to  drop  out  whenever  it  occurs  (see  ex.  4  at  the 
end  of  §  14). 

It  only  remains  to  be  pointed  out  that  these  equalities 
and  those,  which  precede  them,  are  satisfied  by  the  implica- 
tions in  the  table  given  above  and  by  those  contained  in 
the  similar  table  in  §  13. 


CHAPTER   III 

§  21.  In  the  remarks  at  the  end  of  §  8  and  elsewhere  we 
have  referred  to  a  system  of  inference,  in  which  not  all  of 
the  implications  of  the  common  logic  hold  true.  It  is 
proposed  now,  as  a  further  illustration  of  method,  to  con- 
struct in  some  detail  another  system,  some  of  whose  char- 
acteristic postulates  stand  in  contradiction  to  those  of 
§  5  and  §  8.  Without  doing  violence  to  the  fundamental 
conditions  described  in  §  I2  and  §  13,  we  may  assume: 

(i)  a'{aa)  z.  a{aa), 

(ii)  ^'{aa)  z  K^a), 

(iii)  y\ciCi)  z  y(aa), 

(iv)  €  (aa)  z  ^'{aa), 

which  yield  at  once  the  equivalent  set: 

i  z  a{aa), 
i  z  /3(aa), 
i  Z  y{aa), 
e  (aa)   z  0. 

§  22.  In  order  to  frame  an  image  of  the  possibility  of 
a(aa),  fi{aa)  and  y{aa)  standing  for  true  propositions, 
imagine  the  subject-class  and  the  predicate-class,  a  and  b, 
to  approach  connotative  identity.  It  will  then  be  under- 
stood, how  it  might  become  a  question,  as  to  whether 
a{ab),  0(ab)  and  y{ab)  are  to  be  regarded  as  true  or  false 
in  the  limiting  case.  This  image  would,  of  course,  not 
serve  to  guide  us,  unless  the  assumptions  we  have  made 
had  an  analytic  justification.  It  is,  too,  in  a  sense  mis- 
leading, for  we  shall  have  to  conceive  of  (3{ab)  and  y(ab) 
becoming  empirically  untrue  for  special  concrete  meanings 
of  a  and  b  that  render  a{ab)  empirically  true,  without 
making  it  impossible  to  regard  /3(aa)   and  y{aa)  as  true 

22 


Construction  of  a  Non-Aristotelian  Logic      23 

for  all  meanings  of  a.     In  interpreting  ^{aa)  and  y(aa), 
the  part  of  a,  which  is  not  a,  is  taken  to  be  the  null-part. 
§  23.  The  other  postulates,  by  the  aid  of  which  we  shall 
effect  our  solution,  are: 


(v) 

^(ab)  z  Kba), 

(xi) 

{a(ab)   z  Kab)V, 

(vi) 

P{ab)   z  e{ab), 

(xii) 

{a{ab)   z  y(ab)}', 

(vii) 

a(ba)a(bc)  z  c((ca), 

(xiii) 

{y(ab)   z  Kab)V, 

(viii) 

a(ba)  e(bc)  z  e(ca), 

(xiv) 

{Kba)Kcb)  Ze'ica)}', 

(ix)     y{ba)y{cb)  zy(ca),         (xv)      {yiab)y{cb)  Z^'M}', 
(x)     y{ab)€(bc)  z  e{ca),       (xvi)      {(3{ba)y{cb)  /,  e'{ca)\', 
(xvii)      {€(ba)e{cb)   z  e'ica)}'. 

Exercises 

1.  Employing  the  method  of  §  5,  deduce  six  valid  moods  of  the 
form  x{a,  b)  Z  3'(a,  b), 

2.  Deduce  twenty-two  invalid  moods  of  the  form  given  in 
the  last  example. 

Thus,  suppose  y(ab)  z  y(ba)  were  valid, 

{y{ab)  Z  yiba)}   Z  {y{ab)yicb)  Z  y{ba)y{cb)}, 

by  {x  z  y)  Z  (wx  z  wy) ; 

{y{ab)y{cb)  Z  y(ba)y{cb)}{y{ba)y(cb)  Z  e'ica)} 

Z  {y{ab)y{cb)  Z  e'ica)], 

by  ix  Z  y)iy  Z  z)  z  ix  Z  2)  and   a  valid  syllogism  to  be  ob- 
tained later;  but  this  result  is  invalid,  by  (xv); 

.".     yiab)  z  yiba)  is  invalid. 

Again,  suppose  eiab)  Z  ^iab)  were  valid. 

{eiba)  Z  Kba)}   Z  {€iba)aicb)  Z  /3(M«W}, 

by  ix  z  y)  Z.  iwx  z  i^y) ; 

{eiba)oticb)  Z  ^iba)aicb)]{^iba)aicb)  Z  e'ica)] 

Z  {eiba)aicb)  Z  e'ica)], 
by  the  last  result,  a  valid  syllogism  and 

ix  z  y)iy  z  z)  z  ix  z  z). 

Now,  €iba)aicb)  Z  eica)  is  valid  and 

.'.     tiba)aicb)  Z  0, 


24  Non-Aristotelian  Logic 

by  {x  z  y)(x  z  y')  Z  (x  z  o);  and 

t{ba)a{ch)^{cd)  Z  o, 

by  {x  Z  o)  z  {xy  Z  o); 

:.     t{ba)oi{ch)^{cd)  z  ^'(da), 

by  {x  z  o)  z  {x  z  y) ; 

Identifying  c  and  h  and  suppressing  the  a-premise  we  should 
have 

e{ba)l3ibd)  Z  I3'(da), 
which  yields 

^{da)^{bd)  Z  e'(ba); 

but  this  result  contradicts  (xiv) ; 

.*.     e{ab)  Z  ^{ab)  is  invalid. 

3.  Employing  the  method  of  §  8  deduce  eleven  valid  moods  in 
addition  to  (vi)  of  the  form,  x{a,  b)  z  y'{a,  b). 

4.  Deduce  twenty  invalid  moods  of  the  form  given  in  the 
last  example. 

The  only  invalid  mood  of  this  type,  which  offers  any  difficulty, 
is  e(ab)  z  t'{ab).     Suppose,  then,  that  this  mood  were  valid. 

U{ba)  Z  i'iba)}   Z  U(ba)a{cb)  Z  e'{ba)aicb)}, 

by  (x  Z  y)  Z  (wx  z  ivy); 

{eiba)a(cb)  Z  e'{ba)a{cb)}{e'{ba)a{cb)  Z  e(ca)} 

Z  Uiba)aicb)  Z  ^'{ca)}, 

by  (x  Z  y)(y  Z  z)  z  (x  Z  z)  and  a  valid  syllogism  to  be  estab- 
lished later; 

{€{ba)a(cb)  Z  e(ca)}  {e{ba)a{cb)  Z  e{ca)}   Z  {e{ba)a{cb)  Z  0}, 

by  (x  Z  y){x  Z  y')  Z  {x  z  0)  and  a  valid  syllogism  to  be  estab- 
lished later; 

U{ba)a{cb)  Z  0}   Z  [t{ba)a{cb)^{cd)  Z  0], 

by  {x  z  0)  z  {xy  Z  0). 

If,  now,  we  identify  b  and  c  and  suppress  the  a-premise,  we 
should  have  e(ba)0{bd)  Z  0; 

U{ba)^{bd)  Zo\   Z  Hba)^{bd)  Z  ^'{da)\, 

by  {x  z'o)  Z  {x  z  y)\ 

U{ba)^{bd)  Z  ^'(da)}   Z  {I3{da)fi(hd)  Z  e'{ba)} 


Construction  of  a  Non-Aristotelian  Logic      25 

by  {xy  Z  2')  z  (zy  Z  x') ;  but  this  result  contradicts  (xiv)  above, 
and 

.*.     i{ab)  Z  i'iab)  is  invalid. 

5.  Establish  the  invalidity  of  all  the  thirty-two  moods  of  the 
form,  x'{a,  b)  Z  y{a,  b). 

If  we  assume  ^(oi)  Z  0  to  be  true,  the  results  of  the  following 
table  will  then  be  forced  upon  us  by  what  has  gone  before. 
That  is, 

oc'(oo)  Z  0,  a  (oi)  Z  0,  a{io)  Z  0,  a{ii)  Z  0, 

^{00)  Z  0,  )S  {oi)  Z  0,  ^(io)  Z  0,  ^'{ii)  Z  0, 

y'{oo)  Z  0,  Y(oi)  Z  0,  y{io)  Z  0,  y'{n)  Z  0, 

e  {00)  Z  0,  €  {oi)  Z  0,  t{io)  Z  0,  e  (m)  Z  <?, 

and  it  will  be  seen  that  postulate  (xiii)  above  may  now  be  saved. 

6.  Derive  thirteen  valid  moods  of  the  syllogism, 

x{a,  b)y{b,  c)  Z  z{ca), 

as  in  §  16,  from  postulates  (viii-x)  above. 

7.  Prove  that  two  hundred  and  eleven  of  the  invalid  moods 
of  the  type  given  in  the  last  exercise  may  be  shown  to  be  invalid 
by  the  aid  of  the  characteristic  postulates  (i-iv)  above. 

8.  Show  that  the  twenty-eight  invalid  moods  not  accounted 
for  in  the  last  exercise  may  be  made  to  depend  on  one  or  the 
other  of  postulates  (xiv-xvii)  above. 

9.  Derive  the  eighty-one  valid  moods  of  the  syllogism,  x{a,  b) 
y(b,  c)  Z  z'(ca). 

10.  Prove  that  one  hundred  and  forty-four  of  the  invalid  moods 
of  the  type  given  in  the  last  exercise  may  be  reduced  to  invalid 
forms  of  immediate  inference  already  established,  and  so  shown 
to  be  invalid,  by  the  aid  of  the  characteristic  postulates  (i-iv) 
above. 

11.  Show  that  twenty-seven  invalid  moods  not  accounted 
for  in  the  last  exercise  may  be  made  to  depend  on  postulates 
(xiv-xvii)  above. 

12.  As  in  exercise  (i),  §  19,  show  that  no  other  valid  syllogistic 
variations  exist,  except  those  contained  in  the  syllogisms, 
x'(a,  b)y{b,  c)  z  z'(ca)  and  x{a,  b)y'{b,  c)  z  z'{ca). 


CHAPTER    IV 

§  24.  A  sorites  is  a  form  of  implication  of  the  general 
type,* 

x{i,  2)y(2,  3)  "•  z(n  -  I,  n)  z  w{ni). 

Certain  valid  moods  of  the  sorites  can  be  constructed  from 
chains  of  vaHd  syllogisms.     For  example  the  chain, 

y{2i)y{32)   z  yisi), 

7(31)7(43)  z  7(41), 
7(41)7(54)  z  7(51)^ 

will  yield  a  valid  mood,  viz., 

7(21)7(32)7(43)7(54)  z  7(51), 

for 

{7(41)7(54)  I  7(51)]  {7(31)7(43)  ^7(41)] 

Z   {7(31)7(43)7(54)   Z7(5^)}» 
and 

{7(31)7(43)7(54)  ^7(51)]  {7(21)7(32)  A  7(31)] 

z  {7(21)7(32)7(43)7(54)  ^7(51)]- 

Again, 

{a(2l)a(32)   /.  oc(3l)]    Z   {a(2l)a(32)a(43)    £  a(3l)a(43)\. 
.'.     a(2l)a(32)a{43)   z  (^(41). 

The  valid  mood  of  the  sorites  is,   accordingly,   built  up 
out  of  the  chain, 

a(2l)a(32)    Z  a(j7), 
a(3l)a(43)   z  a(4l). 

It  remains  to  be  proven  that  the  only  valid  moods  of  the 
sorites  that  exist  can  be  built  up  from  chains  of  valid  syllogisms. 

It  will  be  convenient  to  take  the  conclusion  successively 
in  each  one  of  the  eight  possible  forms. 

*  In  what  follows  it  will  be  convenient  to  employ  the  ordinal  numbers  for 
class-terms  instead  of  the  initial  letters  of  the  alphabet.  The  solution  given 
here  belongs  to  the  logic  of  the  last  chapter. 

26 


The  General  Solution  of  the  Sorites         27 

Conclusion  a'  and  All  Premises  Unprimed 
Suppose  in  the  first  instance  that  no  e-premise  is  present. 
Then  no  a-premise  is  present,  for  suppose  x{s,  s  -\-  i)  to 
be  an  a-premise.  Identifying  terms  in  the  a-,  fi-  and 
7-premises  except  x,  the  mood  of  the  sorites  will  reduce 
to  an  invahd  mood  of  immediate  inference,  viz., 

a{s,  s  -\-  l)  Z.  oc'{s,  s  +  l). 
Similarly  no  /3-premise  can  occur,  by 

^{s,  s  +  l)  /.  a'{s,  5  +  j), 
and  no  7-premise  can  occur,  by 

y{s,  s  -\-  l)   £  a'(s,  s  +  l). 

Consequently  at  least  one  6-premise  is  present  if  the  mood 
of  the  sorites  is  valid. 

Not  more  than  a  single  e-premise  can  be  present,  for 
if  two  or  more  €-premises  were  present,  the  mood  of  the 
sorites  could,  by  identifying  terms  and  suppressing  the  a-, 
)3-  and  7-premises,  be  reduced  to  the  form, 

e(/,  2)€(2,  j)  •  •  •  e(w  -  J,  n)   Z  (x'{ni), 

and  the  validity  of  this  mood  can  be  made  to  depend  on 
that  of  a  mood  in  which  all  but  two  of  the  premises  are 
absent,*  viz., 

€(^,  5  -f  l)€(s  +  J,  s  +  2)   Z  a'(s  -f  2  s), 

which  is  invalid. 

Contradict  the  e-premise  and  the  conclusion  and  inter- 
change and  the  sorites  reduces  to  the  case  of  an  e'-conclu- 
sion,  which  will  be  considered  later  on. 

*  Thus,  tii2hi23M34hi45)  Z  «'(5^)  yields  e{i4hi43hi34hi45)  L  a'isi) 
for  2=4,  and 

{(ii4M43hi34H45)  Z  "'(5^)1  U(J4)  Z  e(^j)«(j4)) 

Z  Uii4)ei34)<45)  Z  a'isi)]. 
Thus  we  obtain  in  succession, 

t(i4H34h{45)  Z  oc'isi),  for  2  =  4, 
(i34h{45)  Z  oc'isj),  for  1=3. 


28  Non-Aristotelian  Logic 

Conclusion  (3'  or  7'  and  All  Premises  Unprimed 
Exactly  as  in  the  last  case  it  can  be  shown  that  at  least 
one  e-premise  must  be  present  and  that  not  more  than  one 
e-premise  can  occur.  Contradicting  and  interchanging  as 
before  the  e-premise  and  the  conclusion,  the  sorites  reduces 
to  the  case  of  an  e'-conclusion. 

Conclusion  e'  and  All  Premises  Unprimed 
No  e-premise  can  occur,  for,  if  one  or  more  e-premises 
were  present,  the  mood  of  the  sorites  would  reduce  to  the 
form, 

e(j,  2)e(2,  3)  '  •  '  ein  -  i,  n)   z  e'ini), 

bj'  identifying  terms  in  all  of  the  a-,  fi-  and  7-premises, 
and  this  form  is  reducible  to  the  invalid  syllogism, 

e(^,  ^  +  l)€{s  -\-  I,  s  +  2)   Z.  e'(«y  +  2  s), 
or  an  invalid  mood  of  immediate  inference, 

€(^.  s  +  l)   Z  e{s  s  +  7). 

Not  more  than  a  single  i3-premise  can  occur,  for,  if  two 
or  more  /5-premises  were  present,  the  mood  of  the  sorites 
would,  by  identifying  terms  in  all  of  the  premises  but  two 
of  the  j9-premises,  be  reducible  to  an  invalid  syllogism  of 
the  form, 

fi{s,  s  +  l)^(s  -]-  i,s  +  2)   z  e'{s  +  2  s). 

Suppose  that  no  (3-  or  7-premise  is  present.  Then  all  of 
the  premises  are  in  the  a-form  and  the  sorites  becomes, 

a{i,  2)a{2,  3)  •  ■  ■  a{n  -  I,  n)   z  e'(ni), 

which  can  be  constructed  from  the  chain  of  valid  syllogisms, 

a(l,  2)a{2,  3)    Z  a(3l), 
a{3l)a(3,  4)   Z  a(4l), 

a{n  —  2  i)a{n  —  2,  n  —  i)   z  ci{n  —11), 
a{n  —  I  l)a{n  —  I,  n)   Z  e'(w  /). 

If  no  7-premise  is  present  and  all  but  one  /^-premise  is 
in  the  a-form,  the  sorites  then  becomes, . 


The  General  Solution  of  the  Sorites         29 

a(j,  2)a(2,  3)  '•■  a(s  -  I,  s)0(s,  s  +  l)a(s  +  I,  s  +  2) 

'  '  •  a{n  —  I,  n)   z.  ^'{ni), 

which  can  be  built  up  from  the  chain  of  valid  syllogisms, 

a{l,  2)a(2,  3)    Z  aQ/), 
<x(3l)a(3,  4)   Z  a{4l), 

a{s  —  I  l)a{s  —  I,  s)  Z  oi{si), 
a{si)l3{s,  s  -{-  l)  Z  e'(s  +  I  l), 
€'{s  +  I  l)a(s  +  I,  s  +  2)   z  i'(s  +  2  l), 

e'(n  —  I  i)a(?t  —  i,  n)   z  e'(ni). 

Suppose,  again,  that  all  of  the  premises  are  in  the  7-form, 
i.e.,  that  the  sorites  is 

7(7,  2)7(2,  3)  "-  y{n  -  I,  n)   z  e'(wj). 

The  first  premise,  which  presents  the  term-order  (^  —  7  ^), 
i.e.,  with  the  smaller  ordinal  number  appearing  as  subject, 
establishes  that  order  in  each  one  of  the  premises  which 
follow.  For,  suppose  that  the  term  order  (s  —  i  5),  having 
once  occurred,  should  appear  reversed  later  on.  The 
sorites  would,  by  identifying  terms,  be  reducible  to  an 
invalid  syllogism,  viz., 

7(^  —  I  s)y{s  +  I  s)   z  e'(s  +  I  s  —  l). 

The  sorites  becomes,  consequently, 

y{2l)y(32)  •"  y{ss  -  l)y{s  s  +  i) 

•  •  •  y(n  -in)  z  e'(ni), 

which  can  be  derived  from  the  chain, 

y(2i)y(32)   z  y(3i), 

y{3i)y{43)  z  y{4i), 


y{si)y{s  s  -\-  i)  z  e'{s  +  77), 

£'(^  -f  7  7)7(5  +  75  +  2)   z  i'{s  +  2  7), 

t'{n  —  I  i)y{n  —  in)   z  e{ni). 
If  the  mood  of  the  sorites  contains  only  a-  and  7-premises, 


30  Non-Aristotelian  Logic 

the  term  order  in  each  7-premise  is  established  as  above, 
the  first  7-prcmise,  which  presents  the  term  order  (s  —  i  s)' 
establishing  that  order  in  each  7-premise,  which  follows. 
The  generating  chain  of  syllogisms  will  be  the  same  as 
the  last,  except  that  each  minor  premise  in  the  chain, 
which  corresponds  to  an  a-premise  of  the  sorites,  will 
appear  in  the  a-form. 

If  all  of  the  premises,  except  a  single  /3-premise,  be  in 
the  7-form,  the  sorites  becomes, 

7(/,  2)y(2,3)  •••  y{s  -  I,  s)^{s,  s  +  l)y{s  +  i,  s  +  2) 

'  •  •  y{n  —  I,  n)   z  e'{ni). 

The  term-order  in  each  premise,  which  precedes  the  13- 
premise,  is  established  as  (s  s  —  i).  For,  if  the  term-order 
in  a  7-premise  coming  before  the  /3-premise  should  appear 
as  (s  —  I  s),  then,  by  identifying  terms,  the  mood  of  the 
sorites  would  be  reducible  to  an  invalid  syllogism  of  the 
form 

y(s  -  I  s)l3{s,  s  -{-  l)   z  e{s  +  i  s  -  l). 

Each  premise,  which  follows  the  )3-premise,  must  present 
the  term-order  (s  —  i  s),  for  otherwise  the  mood  of  the 
sorites  would  be  reducible  to  an  invalid  syllogism,  vz., 

j8(5  —  2,  s  —  i)y(s  s  —  i)  z  e'(s  s  —  2). 

The  term-order  being  now  unambiguously  established,  the 
sorites  becomes 

y{2i)y{32)  ...  y{s  s  -  i)fi{s,  s  +  7)7(^  -\-  i  s  +  2) 

•  •  •  y{n  —  in)  z  e'{ni), 

which  may  be  generated  from  the  chain, 

y(2i)y(32)  z  7(31), 
7(31)7(43)  /  7(41), 


7(5  -  I  7)7(5  s  -  i)   z  7(si), 

7(si)Ks,  s  +  i)   Z  e(s  +  I  l), 

e'{s  -t-  I  l)7(s  +  I  s  -\-  2)   Z  e'{s  +  2  l), 

t'{n  —  I  i)7(n  —  I  n)  z  ^'(ni), 


The  General  Solution  of  the  Sorites         31 

or,  if  the  initial  premise  be  in  the  /S-form,  from 

I3{l,  2)y{23)   z  €'(J/), 

^'{3i)y{34)  /  ^'{41)^ 


e'in  —  I  i)y(n  —  in)/.  e'{ni). 

The  remaining  moods  of  valid  sorites  which  contain  a-, 
/3-  and  7-premises  and  an  e'-conclusion  are  obtained  from 
the  last  type  by  replacing  one  or  more  7-premises  by  a- 
premises  in  every  possible  way.  Each  type  so  obtained 
can  be  constructed  from  one  of  these  last  chains  of  valid 
syllogisms,  except  that  now  the  minor  premise  of  each 
member  of  the  chain,  that  corresponds  to  an  a-premise 
of  the  sorites,  will  appear  in  the  a-form. 

There  exist,  consequently,  no  valid  moods  of  the  sorites, 
in  which  the  conclusion  is  in  the  e'-form  and  in  which 
none  of  the  premises  is  a  primed  form,  which  cannot  be 
constructed  from  chains  of  valid  syllogisms.  All  the  other 
moods,  in  which  the  premises  are  unprimed  and  the  con- 
clusion is  a  primed  form,  are  gotten  from  the  valid  moods 
already  established  by  the  aid  of  the  principle, 

{xy  A  z)   z.  {xz'  z  3'')- 

§  25.  Conclusion  a  and  All  Premises  Unprimed 

It  will  be  easy  to  show  that  no  ^-,  7-  or  e-premise  can 
occur,  if  the  mood  of  the  sorites  is  valid,  and  that,  conse- 
quently, the  general  form  of  implication  will  be 

ci{i,  2)a(2,  3)  .  .  .  a{n  -  I,  n)   z  oi{ni), 
whose  chain  of  generating  syllogisms  is 
oi{l,  2)a{2,  3)    Z  a{3l), 
oc{3l)oi{3,  4)   Z  «(^^). 

a{:n  —  I  l)a(n  —  i,  n)   Z  cx.{ni). 

Conclusion  jS  and  All  Premises  Unprimed 
Under  this  head  it  will  be  found  that  no  a-,  7-  or  c- 
premise  can  occur  and  that  consequently  all  ot  the  premises 


32  Non-Aristotelian  Logic 

are  in  the  /3-form.     But  such  a  sorites  may  be  reduced  to 
an  invaHd  syllogism,  viz., 

fi(s  -  I,  s)(3is,  s  +  i)  z  Ks  +  I  s  -  i). 

There  exist,  consequently,  no  valid  moods  of  this  type. 

Conclusion  y  and  All  Premises  Unprimed 

Here  it  can  be  shown  at  once  that  no  a-,  ^-  or  c-premise 
can  occur  and  that,  consequently,  all  of  the  premises  are  in 
the  7-form.  Moreover,  the  term-order  in  each  7-premise 
is  established  as  {s  s  —  i),  i.e.,  with  the  larger  ordinal 
number  coming  first;  for,  suppose  one  of  the  premises 
should  appear  as  y{s  —  i  s).  The  mood  of  the  sorites 
would  then  be  reducible  to  an  invalid  syllogism  of  one  of 
the  forms, 

y{s  -  I  s)y(s,  s  +  i)   z  y(s  +  i  s  -  i), 
y{s  —  2,  s  —  i)y{s  —  I  s)   z  y{s  s  —  2). 

The  sorites  becomes,  then, 

y{2i)y{32)  •••  y{nn  -  i)  z  y{n  j), 

and  its  chain  of  generating  syllogisms  is 

y{2i)y{32)  z  y(3i), 

y{3.i)y{43)  L  y{4i), 


y{n  —  I  i)y{n  n  —  i)  z  y{n  i). 

Conclusion  e  and  All  Premises  Unprimed 

Just  as  in  the  cases  already  considered,  it  will  be  easy 
to  show  that  one  and  only  one  e-premise  must  be  present 
and  that  no  /3-premise  can  occur.  One  form  of  this  sorites 
is,  accordingly, 

a(j,  2)a{2,  3)  '"  a(s  -  I,  s)€{s,  s  +  l)a(s  -{-  I,  s  +  2) 

•  •  •  a{n  —  I,  n)   z  i{ni), 

which  can  be  constructed  from  the  chain  of  valid  syllogisms, 


The  General  Solution  of  the  Sorites         33 

a{l,  2)a(2,3)    Z  a(3l), 
oc(3i)a(3,  4)   Z.  oc{4i), 


a{s  —  I  l)oi{s  —  I,  s)   Z  cx(si), 
aisi)€(s,  s  +  l)   Z  e(s  -h  I  l), 
€(s  +  I  l)a{s  -{-  J,  s  +  2)   /.  e{s  +  2  l), 

e(n  —  I  i)a(n  —  i,  n)   z  e{ni). 

The  other  vaHd  sorites  of  this  type  are  gotten  by  re- 
placing one  or  more  a-premises,  coming  before  the  e- 
premise,  by  a  y{s  —  is),  one  or  more  a-premises,  coming 
after  the  e-premlse,  by  a  y{s  s  —  i),  and  it  will  be  easy 
in  each  case  to  construct  the  generating  chain  of  syllogisms. 
There  exist,  consequently,  no  valid  moods  of  the  sorites, 
whose  premises  and  conclusion  are  all  unprimed  forms, 
which  cannot  be  built  up  from  chains  of  valid  syllogisms. 
It  only  remains  to  be  shown  that  the  valid  types  already 
established  are  the  only  valid  types  that  exist,  except 
those  immediately  derived  from  these  by  the  principle, 
(xy  z  z)  z  {xz'  z  y'). 

All  valid  moods  of  the  sorites,  in  which  the  conclusion 
is  a  primed  form  and  a  single  one  of  the  premises  is  a  primed 
form,  follow  at  once  from  the  valid  moods  already  estab- 
lished by  the  aid  of  the  principle  {xy  z  z)  z  {xz'  z  y')- 
For,  suppose  a  valid  mood  of  this  type,  but  not  so  derived, 
should  exist.  Then  a  mood  of  the  sorites  already  estab- 
lished as  invalid  would  appear  as  valid  upon  application 
of   the   same   principle,   i.e.,    {xz'  z  y')  Z  {xy"  Z  z"),   or 

(^3''  ^  2')   Z  (^2  z  3')- 

No  valid  implications  exist,  in  which  a  single  premise  is 
a  primed  form  and  the  other  premises  and  conclusion  are 
unprimed  forms.  Suppose  in  the  first  instance  that  the 
conclusion  is  a  or  7.  Then  if  the  primed  premise  is  e  it 
may  be  strengthened*  to  an  unprimed  /3- premise  and,  if 
the  primed  premise  be  a,  /3  or  7,  it  may  be  strengthened 

*  \i  X  /_  y,  then  x  is  said  to  be  a  strengthened  form  of  y  and  y  is  said  to  be 
a  weakened  form  of  x. 


34  Non-Aristotelian  Logic 

to  €.  In  either  case  the  resulting  mood  is  one  already 
shown  to  be  invalid.  If  the  conclusion  is  13  and  the  primed 
premise  be  strengthened  to  any  unprimed  premise,  the 
resulting  mood  is  invalid.  If,  finally,  the  conclusion  is  e, 
any  primed  e-premise  may  be  strengthened  to  /3,  any 
primed  a-,  /?-,  or  7-premise  to  e,  if  another  €-pemise  is 
present,  and  the  resulting  mood  is  again  invalid.  If  the 
conclusion  is  e  and  no  t-premise  is  present  the  mood  will 
reduce  to  x(s  —  i,  s)  y'{s,  s  -{- 1)  Z  e(s-\-  i  s  —  i).  Continu- 
ing this  same  line  of  reasoning,  it  will  be  seen  that  no  valid 
moods  of  the  sorites  exist,  in  which  the  conclusion  is  un- 
primed and  two  or  more  premises  are  primed. 

Finally  no  valid  moods  of  the  sorites  exist,  in  which  the 
conclusion  is  a  primed  form  and  in  which  two  or  more  of 
the  premises  are  primed  forms.  For  suppose  such  a  mood 
to  exist.  Then,  by  contradicting  and  interchanging  one 
of  the  primed  premises  and  the  conclusion,  the  validity 
of  a  mood  already  found  to  be  invalid  would  follow. 

All  of  the  valid  implications  of  the  general  form, 

.r(j,  2)y{2,  3)  •  "  z(n  -  i,  n)  z  w{ni), 

X  •  •  •  z,  w,  standing  for  either  primed  or  for  unprimed 
letters,  have,  accordingly,  been  established,  without  intro- 
ducing any  assumptions  except  those  essential,  to  the 
solution  of  the  forms  of  immediate  inference  and  of  the 
syllogism.  This  rather  general  type  of  inference  may  be 
expressed  conveniently  in  the  form  of  the  product  of  n 
premises  containing  a  cycle  of  n  terms  and  implying  zero, 
thus: 

x{i,  2)y{2,  3)  ■'•  z{n,  i)  z  0. 

That  the  solution  of  this  last  type  is  exactly  equivalent  to 
the  solution  just  given  follows  from  the  principles, 

{x  z  y)   Z  (xy'  z  o), 
{xy'  L  0)  z  {x  z  y). 

Exercises 

I.  Construct  a  valid  mood  of  the  sorites  from  the  chain  of 
syllogisms, 


The  Geneil\l  Solution  of  the  Sorites         35 

a{2i)y{32)  Z  7(JJ), 
7(j7)a(4j)  Z  y{4l), 

y (41)7(54)  z  7(5 J). 

which  are  valid  in  the  common  logic  (§§  1-19). 

2.  If  €7e  (first  and  second  figure)  and  yet  (second  and  fourth 
figure)  be  regarded  as  invalid  moods  of  the  syllogism  (see  the 
concluding  remarks  of  §  8)  establish  the  invalidity  of  the  sorites, 

y{l2)y(23)  ■•  ■  y(s  -  I  s)e(s,  s  +  7)7(5  +  25  +  /) 

•  •  •  y{n  n  —  i)  Z  ((ni), 
by  the  aid  of  the  following 

Principle. — A  valid  mood  of  the  sorites,  whose  premises  and 
conclusion  are  all  unprimed  forms  and  which  has  one  premise 
of  the  same  form  as  the  conclusion,  will  remain  valid,  when  as 
many  other  /3-  and  7-premises  as  we  desire,  are  put  in  the  a-form. 

3.  Prove,  that  there  exists  no  valid  mood  of  the  sorites,  in 
which  none  of  the  unprimed  premises  has  the  same  form  as  the 
unprimed  conclusion,  by  the  aid  of  the  following 

Principle. — A  valid  mood  of  the  sorites,  whose  premises  and 
conclusion  are  all  unprimed  forms  and  none  of  whose  premises 
has  the  same  form  as  the  conclusion,  will  remain  valid,  when  as 
many  /3-  and  7-premises  as  we  desire  are  put  in  the  a-form. 

4.  Employing  the  principle  of  exercise  2  reduce  the  sorites 
cx{2i)y(j2)a(4j)y(S4)  /_  7(51),  which  is  valid  in  the  common 
logic  (§§  1-19),  successively  to  each  one  of  the  three  valid  syl- 
logisms of  exercise  i . 

5.  Employing  the  same  principle,  establish  the  invalidity  of 
the  sorites, 

y(2i)y(32)y(s4)y(54)  Z  7(5^)- 

6.  From  what  chain  of  valid  syllogisms  (§§  15-19)  can  the 
sorites, 

7(12)7(23)6(3,  4)7(54)7(65)  Z  e(6i), 
be  built  up? 

7.  By  the  aid  of  the  principle  of  exercise  2,  solve  the  sorites 
of  the  common  logic  for  the  case,  in  which  all  of  the  premises 
and  the  conclusion  are  unprimed  forms. 

8.  Complete  the  solution  of  the  sorites  begun  in  exercise  7, 
taking  for  granted  the  following  principles : 

(a)  A  valid  mood  of  the  sorites,  whose  premises  are  all  un- 


36  Non-Aristotelian  Logic 

primed  forms,  whose  conclusion  is  a  primed  form  and  all  of 
whose  premises  and  conclusion  are  of  the  same  form,  will  remain 
valid,  when  as  many  premises,  as  we  desire,  but  one,  are  put 
in  the  a-form, 

(b)  A  valid  mood  of  the  sorites,  whose  premises  are  all  un- 
primed  forms,  whose  conclusion  is  a  primed  form  and  one  of 
whose  premises  is  a  form  different  from  the  conclusion,  will 
remain  valid,  when  as  many  other  jS-  or  7-premises,  as  we  desire, 
are  put  in  the  a-form. 


CHAPTER  V 

§  26.  In  §  12  are  laid  down  certain  conditions,  which 
must  be  taken  account  of  in  setting  down  the  foundations 
of  any  system  of  inference.     The  conditions  are 

a{io)  -\-  y{io)   z  0,         a'{oi)y'{oi)   /.  0,  I 

which  contains  as  a  consequence  a{aa)y'{aa)   z  0,  or  in 
particular, 

a(oo)y'{oo)   z  0,         a{ii)y'{ii)   /.  0.  II 

Thus,  we  should  have  to  have 

{a)  y{oi)  is  a  true  proposition, 

{b)  a{oi),  a{io)  and  y(io)  are  false  propositions, 

(c)  either  a{oo)  or  y{oo)  is  a  true  proposition, 

(d)  either  a{n)  or  y{u)  is  a  true  proposition. 

These  results,  which  are  forced  upon  us  as  a  matter  of 
definition,  leave  us  a  number  of  choices  as  to  the  truth  or 
falsity  of  jS  and  e,  where  subject  and  predicate  are  allowed 
to  take  on  the  meanings  nothing  and  universe  in  every 
possible  way. 

In  order  to  determine  another  of  these  systems,  we 
might,  by  introducing  a  series  of  postulates,  remove  one 
possibility  after  another,  until  no  choice  among  alternatives 
remains.  As  one  further  illustration  of  method,  we  shall 
determine  the  system,  which  appears  the  most  paradoxical 
to  ordinary  intuition,  the  one,  namely,  which  asserts  the 
untruth,  for  all  meanings  of  a,  of  the  proposition  all  a  is 
all  a. 

We  shall  assume  in  the  first  place  that  a(ab),  y{ab)  and 
the  product,  I3'{ab)e'{ab),  are  convertible  by  contraposition, 
i.e.,  denoting  non-a  by  a', 

(l)  fi'(ab)e\ab)   z  fi'{b'a'y{b'a'), 

a(ab)   z  (X  {b'a'), 
y{ab)  z  y  {b'a'). 
37 


38  Non-Aristotelian  Logic 

Our  other  postulates  will  be: 

(2)  a(ab)   z  a'iab')y'(ab'), 

which  yields  a{oo)  z  o,{or  a  =  b  =  o,hy  I.     Consequently, 
y'{oo)  z  0,  by  II;    a(ti)   z  0,  by  (i);    y'iii)   Z  0,  by  II  or 

(I); 

(3)  K^b)   z  a'{ab')y'(ab'), 

which  yields  ^(00)  z  0,  ior  a  =  b  =  0,  by  I;  and  l3{oi)  z  0, 
fi{to)  z  0,  for  a  =  o,  b  =  i,  by  II; 

(4)  ^'{ab)   z  a'iab')y'{ab'), 

which  yields  €'(00)  z  o,for  a  =  b  =  o,hy  I]  and  e'ioi)  z  0, 
^'{io)  z  0,  ior  a  =  b  =  i,  hy  I. 

(5)  a'{ab')y'{ab')   z  e'{ab), 

which  yields  €{u)  z  0,  for  a  =  b  =  i,  by  I. 

The  only  case,  which  remains  unsettled,  is  that  of  ^{ii) 
and  it  may  now  be  seen,  from  the  first  member  of  (i)  that 
(S'(ii)  z  0.  For  convenience  of  reference  we  may  now  sum- 
marize our  results: 

a  (00)  Z  0  OL  {pi)   Z  0  a{io)  Z  0  a  (ii)  Z  0 

/3  {00)  z  o  )8  (oi)   z  0  ^{io)  z  0  ^'{ii)  z  0 

y'{oo)  z  0  y'{oi)  z  0  y{io)  z  0  y'{ii)  Z  0 

i   {00)  Z  0  e'  {oi)  z  0  e{io)  z  0  e  (ii)  z  0 

It  only  remains  to  add  to  what  has  gone  before,  viz., 
a{oo)  z  0,  a{ii)  z  o,  the  more  general  postulate, 

a{aa)   z  0. 

Without  this  postulate  it  still  remains  unsettled,  whether 
we  intend  to  deny,  merely,  the  truth  of  all  a  is  all  a,  or 
to  assert  its  untruth  for  all  meanings  of  a. 

In  the  exercises  below,  it  will  be  taken  for  granted  that 
the  forms  of  immediate  inference,  which  are  untrue  in  the 
common  logic,  are  invalid  in  the  system,  whose  foundations 
are  set  down  here. 


Alternative  Systems  39 

Exercises 

1.  Deduce  the  valid -moods  of  the  syllogism, 

x{a,  b)y(b,  c)  Z  z{ca), 

which  are  twenty-one  in  number,  from  the  following  postulates: 

(i)     a{ha)^{cb)  Z  ^{ca),  (ii)     a{ba)t{ch)  z  e(ca), 

(iii)     y(ba)y(cb)  Z  y{ca),         (iv)     y{ab)t{bc)  Z  ^{ca), 
(v)     ^{ab)  Z  ^{ba),  (vi)     a{ab)  Z  a{ba). 

2.  Deduce  the  valid  moods  of  the  syllogisms,  x'{a,  b)y{b,  c) 
Z  z'{ca)  and  x{a,  b)y'{b,  c)  z  z'(ca),  of  which  there  are  nineteen 
and  twenty-three  respectively,  from  the  results  of  exercise  i. 

3.  Deduce  the  valid  moods  of  the  syllogism,  x{a,  b)y{b,  c) 
Z  z'(,ca),  there  being  one  hundred  and  fourteen  of  this  type,  from 
the  postulates  and  results  of  exercise  i,  by  the  aid  of  the  addi- 
tional postulates: 

(vii)  a{ba)^{cb)  Z  l'{ca), 

(viii)  ^{ba)a{cb)  Z  7'(ca)» 

(ix)  a{ba)  t{cb)  Z  y'{ca), 

(x)  a{ab)  Z  y'{ab), 

(xi)  ^{ab)  Z   e'(a6). 

4.  Show  that  the  members  of  the  following  set  may  be  made 
to  depend  upon  the  implications  that  have  already  been  obtained: 

a{aa)  Z  0L'{aa),  {a{aa)  Z  a{aa)}'y 

maa)  Z  fi'{aa)]\  {^'{aa)  Z  /3  {aa)}', 

{y{aa)  z  y'iaa)}',  y'(aa)  z  7  (aa), 

{  e{aa)  Z   e'(aa)}',  {  e'(oa)  Z   e{aa)}', 

IKab)  Z  y'(ab)V,  {  y(ab)  z  ^'(ab)}', 

{y{ab)  Z   e'(ab)}',  {   e(ab)  Z  y'iab)}'. 

{y{ab)  Z  t'CMK, 

5.  Prove  that  ninety-six  of  the  invalid  moods  of  the  syllogism, 
x{a,  b)y{b,  c)  Z  z'{ca),  may  be  reduced  to  simpler  invalid  forms 
of  inference  already  established  and  so  shown  to  be  invalid, 
(a)  either  by  identifying  terms  in  a  7-premise  or  a  7-conclusion 
and  suppressing  the  part  y{aa),  or,  {b)  by  replacing  the  subject 
and  predicate  of  a  /3-premise  or  a  jS-conclusion  by  unity  and 
suppressing  the  part  j8(m). 

6.  Show  that  the  remaining  forty-six  invalid  moods  not  ac- 


40  Non-Aristotelian  Logic 

counted  for  in  exercise  5  may  be  obtained  by  the  aid  of  the  addi- 
tional postulates: 

(xii)      {a{ba)a(cb)  Z  cx'{ca)}\  (xiii)      {a(ba)  i{cb)  Z   ^'(ca)}\ 

(xiv)      {a(ba)^(cb)  Z  ^'{ca)}',  (xv)      {0(ba)0(cb)  Z  €'(ca)y, 

(xvi)      {a{ba)y(cb)  Z  y'(ca)}',         (xvii)      {y(ba)a(cb)  Z  t'W}'. 

7.  Derive  the  invalidity  of  all  but  eight  of  the  two  hundred  and 
thirty-five  invalid  moods  of  the  syllogism  x{a,  b)y{b,  c)  Z  z{ca) 
from  the  results  established  in  exercises  5  and  6. 

8.  Employ  the  results  of  exercise  7  in  order  to  show  that  there 
exist  no  valid  syllogistic  variations  of  the  form  x'(a,  b)y(b,  c) 

Z  z(ca),    x{a,    b)y'{b,    c)  z  z{ca),    x'{a,    b)y'{b,    c)  z  z(ca),    or 
x'ia,  b)y'{b,  c)  Z  z'{ca). 


14  DAY  USE 

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